Exercise 16, § 4

Let H be a connected closed subgroup of G such that $T\cap H$ is a maximal torus S of H. Let $\lambda \in R(H,S)$; denote by $R(\lambda )$ the set of roots of $R(G,T)$ whose restriction to S is equal to $\lambda$.

a) Show that $R(\lambda )$ is not empty.

b) Let $w\in W_H(S)$; show that there exists an element $\overline{w}$ of $W_G(T)$ such that $R(w\lambda ) = \overline{w}R(\lambda )$ (use Exerc. 4 of §2). c) Let $P(\lambda )$ be the intersection of $R(G,T)$ with the subgroup of X(T) generated by $R(\lambda )$. Show that there exists a closed subgroup $G_{\lambda}$ of G containing T such that $P(\lambda ) = R(G_{\lambda},T)$. Deduce that the reflection $s_{\lambda}\in W_H(S)$ is the restriction to S of a product of reflections $s_{\alpha}$ with $\alpha \in P(\lambda )$.

d) Show that the nodal vector $K_{\lambda}\in L(S)$ associated to $\lambda$ is a linear combination with integer coefficients of the $K_{\alpha}$ for $\alpha \in R(\lambda )$.

e) Let $B_H$ be a basis of $R(H,S)$. Show that $R(\lambda )$ is contained in the subgroup of X(T) generated by the union of the $R(\mu)$ for $\mu\in B_H$ (prove that the set of $\lambda \in R(H,S)$ having the stated property is stable under $s_\mu$ for all $\mu\in B_H$, and use $c$)$)$.